Class SISe_sp

Class to handle the SISe_sp model. This class inherits from SimInf_model, meaning that SISe_sp objects are fully compatible with all generic functions defined for SimInf_model, such as run, plot, trajectory, and prevalence.

Details

The SISe_sp model contains two compartments; number of susceptible (S) and number of infectious (I). Additionally, it contains an environmental compartment to model shedding of a pathogen to the environment. Moreover, it also includes a spatial coupling of the environmental contamination among proximal nodes to capture between-node spread unrelated to moving infected individuals. Consequently, the model has two state transitions,

$$S \stackrel{\upsilon \varphi S}{\longrightarrow} I$$

$$I \stackrel{\gamma I}{\longrightarrow} S$$

where the transition rate per unit of time from susceptible to infected is proportional to the concentration of the environmental contamination $\varphi$ in each node. Moreover, the transition rate from infected to susceptible is the recovery rate $\gamma$, measured per individual and per unit of time. Finally, the environmental infectious pressure in each node is evolved by,

$$\frac{d \varphi_i(t)}{dt} = \frac{\alpha I_{i}(t)}{N_i(t)} + \sum_k{\frac{\varphi_k(t) N_k(t) - \varphi_i(t) N_i(t)}{N_i(t)} \cdot \frac{D}{d_{ik}}} - \beta(t) \varphi_i(t)$$

where $\alpha$ is the average shedding rate of the pathogen to the environment per infected individual and $N = S + I$ the size of the node. Next comes the spatial coupling among proximal nodes, where $D$ is the rate of the local spread and $d_{ik}$ the distance between holdings $i$ and $k$. The seasonal decay and removal of the pathogen is captured by $\beta(t)$. The environmental infectious pressure $\varphi(t)$ in each node is evolved each time unit by the Euler forward method. The value of $\varphi(t)$ is saved at the time-points specified in tspan.

Seasonal Decay ($\beta(t)$): The decay rate $\beta(t)$ is piecewise constant, defined by four intervals determined by the parameters end_t1, end_t2, end_t3, and end_t4 (days of the year, where 0 <= day < 365). The year is divided into four intervals based on the sorted order of these endpoints. The interval that wraps around the year boundary (from the last endpoint to day 365, then from day 0 to the first endpoint) receives the same rate as the interval preceding the first endpoint. Three orderings are supported:

Case 1: end_t1 < end_t2 < end_t3 < end_t4

Interval 1: [0, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3: [end_t2, end_t3) with rate beta_t3
Interval 4: [end_t3, end_t4) with rate beta_t4
Interval 1 (wrap-around): [end_t4, 365) with rate beta_t1

Case 2: end_t3 < end_t4 < end_t1 < end_t2

Interval 3: [0, end_t3) with rate beta_t3
Interval 4: [end_t3, end_t4) with rate beta_t4
Interval 1: [end_t4, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3 (wrap-around): [end_t2, 365) with rate beta_t3

Case 3: end_t4 < end_t1 < end_t2 < end_t3

Interval 4: [0, end_t4) with rate beta_t4
Interval 1: [end_t4, end_t1) with rate beta_t1
Interval 2: [end_t1, end_t2) with rate beta_t2
Interval 3: [end_t2, end_t3) with rate beta_t3
Interval 4 (wrap-around): [end_t3, 365) with rate beta_t4

These different orderings allow the model to handle seasonal patterns where, for example, a winter peak crosses the year boundary.

Details

See also